Discrete Fourier transform: Difference between revisions
julia example
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The inverse transform is given by:
<math>x_n = \sum_{k=0}^{N-1} X_k\cdot e^{i \frac{2 \pi}{N} k n}</math>
=={{header|Julia}}==
<lang julia>function dft(A::AbstractArray{T,N}) where {T,N}
F = zeros(complex(float(T)), size(A)...)
for k in CartesianIndices(F), n in CartesianIndices(A)
F[k] += cispi(-2 * sum(d -> (k[d] - 1) * (n[d] - 1) /
real(eltype(F))(size(A, d)), ntuple(identity, Val{N}()))) * A[n]
end
return F
end
function idft(A::AbstractArray{T,N}) where {T,N}
F = zeros(complex(float(T)), size(A)...)
for k in CartesianIndices(F), n in CartesianIndices(A)
F[k] += cispi(2 * sum(d -> (k[d] - 1) * (n[d] - 1) /
real(eltype(F))(size(A, d)), ntuple(identity, Val{N}()))) * A[n]
end
return F ./ length(A)
end
const seq = [2, 3, 5, 7, 11]
const fseq = dft(seq)
const newseq = idft(fseq)
println("$seq =>\n$fseq =>\n$newseq =>\n", Int.(round.(newseq)))
</lang>{{out}}
<pre>
[2, 3, 5, 7, 11] =>
ComplexF64[28.0 + 0.0im, -3.3819660112501033 + 8.784022634946172im, -5.618033988749888 + 2.800168985749483im, -5.618033988749888 - 2.800168985749483im, -3.381966011250112 - 8.78402263494618im] =>
ComplexF64[2.0000000000000013 - 1.4210854715202005e-15im, 2.999999999999996 + 7.993605777301127e-16im, 5.000000000000002 + 2.1316282072803005e-15im, 6.999999999999998 - 8.881784197001252e-16im, 11.0 + 0.0im] =>
[2, 3, 5, 7, 11]
</pre>
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